1. In the navier stokes equation we have the term ([itex]\vec{u}[/itex] [itex]\bullet[/itex]∇)[itex]\vec{u}[/itex](adsbygoogle = window.adsbygoogle || []).push({});

If I have [itex]\vec{u}[/itex] = f(r)(-y,x) with r= [itex]\sqrt{x^2+y^2}[/itex] then is there some some of product rule/identity that needs to be invoked for the initial dot product?

I would say this calculation is:

(-y*f(r)[itex]\frac{\partial}{\partial x}[/itex] + x*f(r)[itex]\frac{\partial}{\partial y}[/itex]) (-y*f(r),x*f(r))

Then:

(-y*f(r)*(0-y*[itex]\frac{\partial f}{\partial r}[/itex]*[itex]\frac{\partial r}{\partial x}[/itex]) + x*f(r)*(-f(r)-y*[itex]\frac{\partial f}{\partial r}[/itex]*[itex]\frac{\partial r}{\partial y}[/itex]), 2nd component)

Please confirm.

Also, typically we consider the dot product to be commutative. But this instance is not commutative, as far as I can see. Is this because it's not considered 'true' usage of the dot product, since the del operator is not technically a vector?

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# 2 questions about dot product

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